Elasticity
modulus,
shrinkage,
and creep of
high-strength
concrete
as adopted
by aashto
Nabil Al-Omaishi,
Maher K. Tadros,
and Stephen J. Seguirant
In the past 20 years, the use of high-strength concrete
(HSC) to improve the structural efficiency of pretensioned
concrete girders has increased significantly. It is now
standard practice to specify design concrete compressive
strengths in excess of 8 ksi (55 MPa). In many regions,
specifying 10 ksi to 12 ksi (69 MPa to 83 MPa) compressive-strength concrete results in little, if any, increase in
girder cost compared with the standard 6 ksi (41 MPa)
concrete used prior to the early 1990s.
Editor’s quick points
n Designers, owners, and precasters use high-strength concrete
(HSC) because it can increase spans and spacing distances of
members and reduce member section sizes.
n HSC’s long-term material properties cannot be accurately
extrapolated from those of normal-strength concrete.
n This paper includes results of research focused on the longterm material properties of HSC.
44
S um me r 2 0 0 9 | PCI Journal
HSC allows the use of greater levels of prestressing, thus
increasing member span and spacing capabilities. Extrapolating the material property and prestress loss prediction
methods developed for 4 ksi to 6 ksi (28 MPa to 41 MPa)
concrete strengths to HSC has resulted in unrealistically
high prestress loss estimates1 and inaccurate camber and
deflection predictions. A recent independent study by
Stallings et al.2 has confirmed that the pre-2005 American
Association of State Highway and Transportation Officials
(AASHTO) LRFD Bridge Design Specifications3 formulas
for predicting long-term concrete material properties do
not provide reliable estimates for HSC. There is a need for
more accurate methods to estimate the material properties
of HSC.
This paper covers the experimental and theoretical components of National Cooperative Highway Research Program
(NCHRP) research project no. 18-07, which is discussed
extensively in NCHRP report 496.4 These components are
0.160
Unit weight wc, kip/ft 3
0.155
0.150
wc = 0.001 f'c + 0.1373
0.145
0.140
Test data
0.135
Linear (t est data)
0.130
4
6
8
10
12
14
16
18
20
Compressive strength f'c , ksi
Figure 1. This graph shows the relationship between the density and compressive strength of concrete. Note: 1 ksi = 6.895 MPa; 1 kip/ft3 = 0.016 kg/m3.
related to modulus of elasticity, shrinkage, and creep of
concrete. The experimental program was conducted at various bridge sites and at the University of Nebraska–Lincoln
(UNL) for specimens produced from raw materials and
mixture proportions provided by four participating states:
Nebraska, New Hampshire, Texas, and Washington. These
locations were selected for their geographic diversity to
have a valid representation of U.S. materials and weather
conditions. Previously reported measurements of material
properties are also included.
shrinkage, and creep of HSC. To help with clarity, the
notation and units employed in the pre-2005 AASHTO
LRFD specifications will be adopted as much as possible.
Stresses will be expressed in units of ksi (MPa) rather than
psi, as is generally used in Building Code Requirements for
Structural Concrete (ACI-318-99) and Commentary (ACI
318R-99).6
The experimental program was used to extend the pre-2005
AASHTO LRFD specifications prediction formulas to concrete with compressive strengths up to 15 ksi (104 MPa).
For each material property, a summary of the experimental
values is presented followed by a comparison with the values obtained from the pre-2005 AASHTO LRFD specifications and the American Concrete Institute (ACI) 209 committee report.5 The proposed formulas provide designers of
prestressed concrete girders with more realistic estimates
of long-term material properties, including effects of aggregate type and other significant factors. The use of the
proposed formulas should give results comparable to those
using the pre-2005 AASHTO LRFD specifications when
concrete compressive strengths are close to 4 ksi (28 MPa).
The use of these formulas with higher-strength concrete
should result in more realistic camber predictions and
lower prestress loss estimates.
Modulus of elasticity
The following sections present the background and recommendations for prediction of the modulus of elasticity,
Influencing factors and
pre-2005 AASHTO prediction
In North American practice, the modulus of elasticity
of concrete has traditionally been considered to increase
approximately with the square root of the compressive
strength. Also, the modulus of elasticity has traditionally
been assumed to vary with the density of concrete raised to
the power of 1.5. This tradition was followed in this study.
Equation (1) is the calculation for the modulus of elasticity
of concrete Ec according to the pre-2005 AASHTO LRFD
specifications Eq. (5.4.2.4.1):
Ec = 33,000w1.5
f c' c
(1)
where
f c' = compressive strength of concrete
wc = density of concrete
PCI Journal | S u m m e r 2009
45
NCHRP report 496 has an equivalent equation in metric.
The equation in section 8.5.1 of ACI 318 is identical to Eq.
(1) except that the units are based on pounds rather than
kips. The data used to develop this equation were based
on concrete strengths ranging from 2 ksi to 7 ksi (14 MPa
to 48 MPa). The data with relatively weak aggregate were
omitted. The density of concrete ranged from 0.08 kip/ft3
to 0.150 kip/ft3 (1280 kg/m3 to 2400 kg/m3).
The ACI 363 committee report7 indicates that Eq. (1) may
overestimate the modulus of elasticity for compressive
strengths over 6 ksi (41 MPa). That position was primarily
based on the work of Carrasquillo.8 The committee report
recommends that the modulus of elasticity be estimated
using Eq. (2).
(
Ec = wc / 0.145
)1.5 ⎛⎝ 1000 + 1265
f c' ⎞ ⎠
(2)
surface ratio, ambient relative humidity, concrete age, type
of curing, and age of concrete under service. It is conveniently expressed as a dimensionless strain under uniform
conditions of relative humidity and temperature. The pre2005 AASHTO LRFD specifications provided formulas
for estimating shrinkage.
For accelerated curing, shrinkage strain εsh is calculated
from Eq. (4).
εsh = (560 × 10-6)ktd ks khs (4)
where
ktd = time-development factor
ks = size factor for the effect of the volume-to-surface
ratio for shrinkage
As will be seen from the correlation with test results, the
authors have not detected any improvement in predicting
Ec with Eq. (2). The research on which this paper is based
was limited to normalweight concrete.
khs = humidity factor for shrinkage
Many designers commonly use default values for the density of normalweight aggregate concrete when estimating
Ec. These are generally assumed as 0.145 kip/ft3
(2320 kg/m3) for cast-in-place concrete and 0.150 kip/ft3
(2400 kg/m3) for precast concrete. However, concrete with
relatively high strength has a low water-cement ratio and a
relatively high density. Russell9 has developed a best-fit relationship between density and strength (Fig. 1). The data
further indicate that nearly all mixtures had a density less
than 0.155 kip/ft3 (2480 kg/m3). This was later confirmed
through a survey of the concrete producers in areas where
dense aggregates are used. A simplified version of Russell’s relationship can be used to represent the data in Fig.
1 with an upper limit of 0.155 kip/ft3 (2480 kg/m3) and a
lower limit of 0.145 kip/ft3 (2320 kg/m3).
wc = 0.140 +
f c'
1000
For moist curing, shrinkage strain εsh is calculated from
Eq. (5).
εsh = (510 × 10-6)ktd ks khs
After one day to three days of accelerated curing, the timedevelopment factor for shrinkage ktd is determined by Eq.
(6).
ktd =
t
Shrinkage and creep
Shrinkage is influenced by factors such as volume-to-
46
S um me r 2 0 0 9 | PCI Journal
(6)
= drying time after end of curing, days
After seven days of moist curing, the ktd and ks are determined by Eq. (7) and (8), respectively.
ktd =
where
Equations (1) and (2) do not account for the effect of
aggregate type. It has been observed10,11 that stiff coarse
aggregates can produce significantly higher modulus of
elasticity for concretes of the same strength and density.
As a result, the experimental work reported in this paper
included identification of aggregate types and sources.
t
55 + t
where
(3)
0.145 ≤ wc ≤ 0.155
(5)
t
0.36V/ S
ks = 26e
t
45 + t
+t
t
35 + t
1064 94V/ S
923
(7)
(8)
where
V/S= volume-to-surface ratio of the exposed surfaces of
the component
For average annual ambient mean relative humidity RH
less than 80%, the humidity factor for shrinkage khs is
calculated from Eq. (9).
khs =
140 − RH
70
(9)
For RH greater than or equal to 80%, the humidity factor
for shrinkage khs is calculated from Eq. (10).
khs =
(
3 100 − RH
)
(10)
not intended to be used for excessively high compressive
stress. Structural analysis modeling allows use of the creep
coefficient for cases where the stress in concrete varies
with time, such as in the case of prestress losses and with
deck or girder differential creep and shrinkage. Equations
(11) through (16) are the pre-2005 AASHTO LRFD specifications creep-prediction formulas.
ψ(t,ti) = 3.5kf kc khc kla ktd
(11)
where
70
The creep coefficient ψ(t,ti) is the ratio of creep strain occurring in the period t to the elastic strain at ti caused by a
constant stress applied to concrete of age ti and sustained
in the period t, where t is the age of concrete between time
of loading for creep calculations, end of curing for shrinkage calculations, and time being considered for analysis of
creep or shrinkage effects and ti is the age of concrete when
load is initially applied. Creep strain will reach its ultimate
value at the end of the service life of the structure. The
creep coefficient is influenced by the same factors that influence shrinkage as well as the age of concrete at the time
of loading. The coefficient is defined in such a way that the
applied stress has to be a constant sustained stress within
the levels that usually prevail for in-service conditions. It is
kf = concrete strength factor =
1
f'
0.67 + c
9
(12)
kc = size factor for creep
t
=
26e
0.36V/ S
t
45 + t
+t
1.80 1.77e 0.54V/ S
2.587
(13)
khc = humidity factor for creep
khc = 1.58 −
RH
120
(14)
Table 1. Laboratory and field materials testing program
Number of specimens
Testing
Concrete age, days
Deck
Girder
Field
4 ksi
8 ksi
10 ksi
12 ksi
10 ksi
f c' and Ec
1
3
3
3
3
3
f c' and Ec
3
3
3
3
3
3
f c' and Ec
7
3
3
3
3
3
f c' and Ec
14
3
3
3
3
3
f c' and Ec
28
3
3
3
3
3
f c' and Ec
56
3
3
3
3
3
f c' and Ec
90
3
3
3
3
n.d.
f c' and Ec
128
3
3
3
3
n.d.
f c' and Ec
256
3
3
3
3
n.d.
7-day moist curing
3
n.d.
n.d.
n.d.
n.d.
1-day accelerated curing
n.d.
3
3
3
3
1-day loading
n.d.
3
3
3
n.d.
56-day loading
n.d.
3
3
3
n.d.
Shrinkage
Creep
Note: Ec = modulus of elasticity of concrete; f c' = specified compressive strength of concrete at 28 days unless another age is specified. n.d. = no data.
1 ksi = 6.895 MPa.
PCI Journal | S u m m e r 2009
47
−0.118
kla = loading age factor = ti
ktd =
(15)
(t − ti )0.6 0.6
10 + (t − ti )
(16)
experimental program
The experimental program consisted of materials testing
in the laboratory and in the field, as well as testing of fullscale, high-strength prestressed concrete bridge girders in
Nebraska, New Hampshire, Texas, and Washington. The
following discussion is limited to a description of specimens used for evaluation of modulus of elasticity, creep,
and shrinkage. Details of the girder testing will be covered
in a subsequent paper.
The material testing program consisted of laboratory tests
conducted at UNL and field tests conducted at production
plants and construction sites. The concrete for each state
included three HSC girder mixtures with design compressive strengths ranging from 8 ksi to 12 ksi (55 MPa to
83 MPa) and one normal-strength deck concrete with design compressive strength of 4 ksi (28 MPa). The precast
concrete producer in each of the four states provided the
mixture proportions and raw materials for production and
testing of the specimens at UNL. In addition, each participating state highway agency provided the raw material and
the mixture proportions for the deck concrete.
Specimens for testing compressive strength and modulus of elasticity were 4 in. × 8 in. (100 mm × 200 mm)
cylinders. Creep and shrinkage specimens were 4 in. ×
4 in. × 24 in. (100 mm × 100 mm × 600 mm) prisms. The
concrete cylinders were made in accordance with ASTM
Table 2. Concrete mixture proportions
Mixture
designation
Coarse aggregates
Type
Weight,
lb
Size, in.
NE04D
Limestone
1.50
NE09G
Limestone
NE10G
Fine aggregates
Type
Cement
Fly ash
Weight,
lb
Weight,
lb
Type
Weight,
lb
Class
Weight,
lb
Air, %
883
Sand/gravel
2039
263
I
658
n.a.
n.a.
6
0.75
1530
Sand/gravel
1530
250
III
705
n.a.
n.a.
5-7
Limestone
0.50
1860
Sand
990
240
I
750
C
200
5-7
NE12G
Limestone
0.375
1913
Sand/gravel
933
254
III
680
C
320
5-7
NE field
Limestone
0.75
1530
Sand/gravel
1530
250
III
705
n.a.
n.a.
5-7
NH04D
Gravel
1.00
1805
Sand
1205
250
II
658
F
132
2
NH10G
Gravel
0.75
1850
Sand
940
250
II
800
n.a.
n.a.
2
NH11G
Gravel
0.75
1850
Sand
925
250
II
800
n.a.
n.a.
2
NH12G
Gravel
0.75
1850
Sand
950
242
II
800
n.a.
n.a.
2
NH Field
Gravel
0.75
1850
Sand
940
250
II
800
n.a.
n.a.
2
TX04D
Gravel
0.75
1811
Sand/gravel
1192
244
I
611
C
152
2
TX08G
Limestone
0.75
2029
Sand
1237
206
III
611
n.a.
n.a.
2
TX09G
Limestone
0.75
2011
Sand
1340
192
III
564
n.a.
n.a.
2
TX10G
Limestone
0.75
1975
Sand
1237
197
III
705
n.a.
n.a.
2
TX field
Limestone
0.75
2011
Sand
1340
192
III
564
n.a.
n.a.
2
WA04D
Gravel
1.00
1810
Sand
1046
263
I
660
F
75
2
WA10G
Gravel
0.75
2010
Sand
1235
219
III
705
n.a.
n.a.
1.5
WA11G
Gravel
0.50
1877
Sand
1383
217
III
658
n.a.
n.a.
1.5
WA12G
Gravel
0.375
1959
Sand
1204
213
III
752
n.a.
n.a.
1.5
WA field
Gravel
0.75
2010
Sand
1235
219
III
705
n.a.
n.a.
1.5
Note: n.a. = not applicable. 1 in. = 25.4 mm; 1 lb = 0.453 kg.
48
Water
S um me r 2 0 0 9 | PCI Journal
NCHRP 496 test data
AASHTO LRFD and ACI 318 (Eq. [1])
12,000
ACI 363 (Eq. [2])
Proposed (Eq. [3])
Modulus of elasticity Ec, ksi
10,000
8000
6000
4000
2000
0
0.000
2.000
4.000
6.000
8.000
10.000 12.000
Compressive strength f'c, ksi
14.000
16.000
18.000
20.000
Figure 2. The modulus of elasticity was determined using results of experiments in this project. Note: AASHTO = American Association of State Highway and Transportation
Officials; ACI = American Concrete Institute; LRFD = load- and resistance-factor design; NCHRP = National Cooperative Highway Research Program. 1 ksi = 6.895 MPa.
C19212 and were cured in the laboratory curing room at
an ambient temperature of 73 oF (23 oC) for 24 hr. Table
1 summarizes the laboratory and field testing program.
Table 2 gives the mixture proportions of each concrete
designation.
The testing for compressive strength, modulus of elasticity,
shrinkage, and creep was performed according to ASTM
C39,13 ASTM C469,14 modified ASTM C157,15 and ASTM
C512,16 respectively.
All modulus-of-elasticity data were based on membercured cylinders for field tests and moist-cured cylinders for
laboratory tests. The on-site cured cylinders were subjected
to the same long-term curing and storing conditions as
those of the actual members, in accordance with ASTM
C1231.17
The shrinkage specimens were cast at the same time and
cured under the same conditions as the creep specimens.
Readings were taken in parallel with the creep tests for
each mixture to compare the time-dependent strain of
loaded and unloaded specimens. The creep and shrinkage
specimens in this project had a V/S of 1.0. The specimens
were stored at an ambient RH of 35% to 40%.
Demountable mechanical (DEMEC) gauges were used
to measure the surface strains in the longitudinal direc-
tion. Five DEMEC points were used on each of the two
opposite faces of the specimens and were spaced at 4 in.
(100 mm). This allowed for three 8 in. (200 mm) gauge
lengths per surface, or six readings per specimen. Shrinkage readings were taken daily for the first week, weekly
for the first month, and monthly for about a year.
Creep tests were performed in the laboratory on the twelve
HSC mixtures. Similar to the shrinkage strain measurements, DEMEC gauges were used. A total of four specimens were made for each mixture. Three of these specimens were loaded at the age of one day, while the fourth
was loaded at the age of fifty-six days. The specimens
were loaded with an intensity of not more than 40% of the
concrete compressive strength at the age of loading. The
loading was initially applied using a hydraulic jack and
measured with a load cell. Through nut tightening, the
load was then transferred from the jack to the compressed
spring. The level of sustained stress was kept constant
through frequent measurements and adjustments.
The initial strain readings were taken immediately before
and after loading. Creep measurements were then taken
daily for the first week, weekly for the first month, and
monthly for about a year. The creep coefficients were
calculated from the measured total strains, elastic strains,
and shrinkage strains.
PCI Journal | S u m m e r 2009
49
ACI 363 test data
FHWA test data
12,000
NCHRP 496 test data
AASHTO LRFD and ACI 318 (Eq. [1])
ACI 363 (Eq. [2])
Modulus of elasticity Ec, ksi
10,000
Proposed NCHRP 496 (Eq. [3])
8000
6000
4000
2000
0
0.000
2.000
4.000
6.000
8.000
10.000
12.000
Compressive strength f'c, ksi
14.000
16.000
18.000
20.000
Figure 3. This graph shows the modulus of elasticity, including results of previous research. Note: AASHTO = American Association of State Highway and Transportation
Officials; ACI = American Concrete Institute; FHWA = Federal Highway Administration; LRFD = load- and resistance-factor design; NCHRP = National Cooperative Highway
Research Program. 1 ksi = 6.895 MPa.
Proposed prediction methods
where
The proposed prediction methods are presented as adopted
by the AASHTO LRFD interim specifications for 200518
and 2006.19 The same provisions appear in the fourth edition
in 2007.20 Slight modifications were made by the AASHTO
subcommittee T10, Concrete Bridges, to the original proposal presented by the authors in NCHRP report 496. These
modifications will be summarized in the following sections.
f c' = specified concrete compressive strength at service
Proposed modulus-of-elasticity
formula
There was considerable scatter in the modulus-of-elasticity
data (Fig. 2 and 3). Introducing a variable density with concrete strength and an aggregate stiffness factor K1 provides
improvement to the results for HSC with unusually stiff or
soft aggregates. Equation (17) is the proposed formula.
Ec = 33,000K1w1.5
f c' c
Proposed shrinkageand creep-prediction formulas
Equations (18) and (19) are intended to represent the test
data with a rectangular hyperbolic equation, similar to that
in the ACI 209 committee report and the pre-2005
AASHTO LRFD specifications but with modifications to
account for the effects of HSC.
where
K1 = 1.0 unless determined by physical test and as approved by the authority of jurisdiction
where
wc = (0.140 + f c' /1000)
and 0.145 kip/ft3 < wc < 0.155 kip/ft3
50
(17)
The density of concrete wc is assumed not to vary with
time by taking f c' constant. This is an improvement over
the original proposal, where wc varied with concrete
strength as concrete aged. However, f c' in Eq. (17) is variable with time. It is compressive strength of concrete at the
same concrete age at which the modulus of elasticity is to
be determined.
S um me r 2 0 0 9 | PCI Journal
εsh = (480 × 10-6)ktd kvs kf khs
(18)
kvs = a factor for the effect of volume-to-surface ratio
( t,ti ) = 1.90ktd kvs k f khcti 0.118 (19)
The ultimate creep coefficient was set at 1.90 for average
conditions. This definition differs from that of the pre-2005
AASHTO LRFD specifications in which the ultimate creep
coefficient was 3.5 for standard conditions. The difference
between average and standard conditions will be discussed
in the next section.
For example, average RH is 70% in the new provisions
and standard RH is 40% in the pre-2005 AASHTO LRFD
specifications. The correction factors in the new provisions are equal to unity for average conditions, while they
were set equal to unity in the pre-2005 AASHTO LRFD
specifications for standard conditions. A similar strategy
was used to establish the ultimate shrinkage strain of 480 ×
10-6 to represent the ultimate strain at average conditions.
It is somewhat different from the values shown in Eq. (4)
and (5) of 560 × 10-6 and 510 × 10-6 for accelerated and
moist curing under standard conditions. Both the creep and
shrinkage formulas yield results comparable to those of
the pre-2005 AASHTO LRFD specifications for concrete
strength at prestress transfer of 4.0 ksi (28 MPa), assumed
in this paper to be equal to about 5.0 ksi (35 MPa) at 28
days, if other influencing factors are unchanged.
Proposed correction factors
for shrinkage and creep
under nonstandard conditions
Correction factors were used in the pre-2005 AASHTO
LRFD specifications methods to modify the values of ultimate shrinkage and creep for any periods shorter than full
service life and for nonstandard conditions. These standard
conditions, in some methods, referred to laboratory specimen sizes and environmental conditions. For example,
the ACI 209 committee-report method and the pre-2005
AASHTO LRFD specifications shrinkage-prediction methods consider an RH of 40% to be a standard condition,
while most U.S. bridges are subjected to an average RH
of about 70%. Also, the standard V/S was taken as 1.5 in.
(38 mm) in the pre-2005 AASHTO LRFD specifications,
while the average for most bridge members is about 3.5 in.
(89 mm). The following correction factors have been reformatted to be equal to unity under average conditions.
Ambient relative humidity
correction factor
Equations (20) and (21) are simplifications of the pre-2005
AASHTO LRFD specifications equations for shrinkage (Eq. [9] and [10]) and for creep (Eq. [14]). Figure 4
shows a comparison of the various prediction methods
normalized to unity at an RH of 70%. This figure shows
two trends when normalized to a default value of 1.0 at
1.80
Shrinkage
ACI 209: 1.43(1.4 - 0.01RH) for RH < 80
ACI 209: 1.43(3.00 - 0.03RH) for RH > 80
AASHTO LRFD specifications: (140 - RH)/70 for RH < 80
AASHTO LRFD specifications: 3(100 - RH)/70 for RH > 80
PCI bridge design manual: khs = 2.00 - 0.0143RH for RH < 80
PCI bridge design manual: khs = 4.286 - 0.0429RH for RH > 80
Humidity correction factors khc and khs
1.60
1.40
1.20
1.00
Creep
ACI 209: 1.25(1.27 - 0.0067RH)
Pre-2005 AASHTO LRFD specifications: khc = 1.58 - RH/120
PCI bridge design manual: khc = 1.586 - 0.0084RH
0.80
0.60
0.40
Practical range
0.20
0.00
30
40
50
60
70
80
90
Relative humidity RH, %
Figure 4. This graph determines the humidity correction factor according to various prediction methods. Note: Values predicted by various methods are normalized to unity
at a humidity of 70%. AASHTO = American Association of State Highway and Transportation Officials; ACI = American Concrete Institute; LRFD = load- and resistance-factor
design.
PCI Journal | S u m m e r 2009
51
1.60
Shrinkage
ACI 209, PCI bridge design manual: 1.25 ks = 1.5e-0.12V/S
1.50
AASHTO LRFD specifications: 1.25ks = (1064 - 94V/S)/735
1.40
Creep
ACI 209, PCI bridge design manual:
1.25ks = 2.5/3 (1 + 1.13e-0.54V/S)
Size factor ks
1.30
AASHTO LRFD specifications:
1.25ks = (1.25)(1.80 + 1.77e-0.54V/S)/2.587
1.20
1.10
1.00
0.90
0.80
Laboratory
specimen
range
Precast concrete
stemmed members
0.70
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
4.50
Volume-to-surface ratio V/S, in.
Figure 5. This graph determines the size correction factor according to various methods. Note: AASHTO = American Association of State Highway and Transportation Officials; ACI = American Concrete Institute; LRFD = load- and resistance-factor design. 1 in. = 25.4 mm.
70% RH. Because the great majority of applications fall
in the range of 30% to 80% ambient RH, the relatively
low shrinkage coefficient for humidity higher than 80%
is proposed to be conservatively ignored. This allows for
reduction of the correction factor to just one formula for
shrinkage (Eq. [20]) and another for creep (Eq. [21]).
khs = 2.00 – 0.014RH
(20)
khc = 1.56 – 0.008RH
(21)
The three formulas produce close results when used for the
common range of V/S. Thus, it is proposed to use the simplest of the formulas (Eq. [8]) with the first bracketed term
reduced to 1 due to the time t being taken equal to infinity:
kvs =
1064 94V/ S
735
= 1.45 – 0.13(V/S) ≥ 0
(22)
A lower limit of zero must be placed on kvs to eliminate
the possibility of irrationally using a negative shrinkage or
creep for relatively thick members.
Loading-age correction factor
Size correction factor
Relatively thick members do not dry as rapidly as thin
members when subjected to ambient air with humidity less
than 100%. This effect is captured by using the V/S factor.
Member size affects short-term creep and shrinkage more
than it does the ultimate values. The ultimate values are the
ones of primary importance for stringer-type bridges. The
size-factor formula is proposed to be simplified by using
a time duration equal to infinity. Figure 5 shows a comparison of the correction factors according to the pre-2005
AASHTO LRFD specifications, PCI's Precast Prestressed
Concrete Bridge Design Manual,21 and the ACI 209 committee-report formulas normalized for V/S equal to 3.5 in.
(89 mm). This ratio corresponds to that for an I-girder with
a web width of about 7 in. (180 mm).
The pre-2005 AASHTO LRFD specifications and the ACI
209 committee-report prediction formulas were examined
for computing the loading-age correction factor for both
accelerated and moist curing conditions. Figure 6 presents
the correction factor for a range of loading ages normalized to a value of 1.0 for one day of accelerated curing or
seven days of moist curing. This figure indicates that the
variation of the correction factor with loading age follows
a similar trend for both types of curing. Thus, the pre-2005
AASHTO LRFD specifications formula should continue to
be used for both types of curing, with a shift in datum used
to represent the difference in curing type. Accordingly, Eq.
(15) is proposed for calculating the loading-age correction
factor kla.
52
S um me r 2 0 0 9 | PCI Journal
kla = ti−0.118 (15)
1.40
ACI 209: kla = 1.13tla-0.094 accelerated curing
1.20
Loading correction factor kla
ACI 209: kla = 1.25tla-0.118 moist curing
1.00
0.80
0.60
AASHTO LRFD specifications: kla = ti-0.118 Eq. (15) accelerated or moist curing
0.40
0.20
0.00
0
7
14
21
28
35
42
49
56
Loading age ti, day
Figure 6. This graph determines the loading-age correction factor according to various methods. Note: AASHTO = American Association of State Highway and Transportation Officials; ACI = American Concrete Institute; LRFD = load- and resistance-factor design; tla = .
It is assumed that moist-cured concrete reaches the same
level of maturity at seven days that accelerated-cured
concrete reaches in one day. Thus, ti is to be taken as equal
to the actual concrete age for accelerated curing and the
concrete age at the time of loading minus six days for
moist-cured concrete loaded after a minimum of seven
days of moist curing. Precast, prestressed concrete girders
are generally assumed to have the first loading application
at one day. That loading consists of the initial prestressing plus self-weight. Deck slabs that are made composite
with the girders are assumed in the analysis to begin to
interact with the girders after seven days of curing, creating
differential shrinkage and creep. Additional load applications on the girder, namely deck weight and superimposed
dead loads due to barriers and wearing surface, should be
analyzed with ti values corresponding to the actual age of
the girder concrete.
Strength correction factor
The strength correction factor is one of the primary changes introduced in the new provisions. The ACI 209 and
the pre-2005 AASHTO LRFD specifications shrinkageprediction methods do not include a correction factor for
concrete strength. The experimental results in this research
clearly show the impact of HSC on reducing both creep
and shrinkage. Figure 7 shows a comparison of the correction factors according to the pre-2005 AASHTO LRFD
specifications creep factor, Al-Omaishi,22 and the proposed
factor. Al-Omaishi demonstrated that creep and shrinkage
should be more accurately related to concrete strength at
the time of prestress release fci' than to the compressive
strength at 28 days or 56 days.
The concrete strength factor obtained with the pre-2005
AASHTO LRFD specifications formula was normalized
to a value of 1.0 for a final compressive strength in
service f c' of 5.0 ksi (35 MPa), with the assumed relationship f c' = ( fci' /0.8). This assumption would allow usage of
the same formulas in estimating creep and shrinkage of the
deck slab, which has much less of an impact on the overall
prestress loss and deformation of the bridge superstructure than does that of the girders. Therefore, the strength
correction factor for both shrinkage and creep of concrete
may be computed from Eq. (23).
kf =
5
1+ f ci'
(23)
For nonprestressed members, fci' may be taken as 0.80 f c' .
Time-development correction factor
The time-development correction factor is used to estimate
creep and shrinkage effects at times other than infinity.
These effects are important in bridge design and construction if a relatively accurate camber prediction at the time
PCI Journal | S u m m e r 2009
53
1.60
Strength correction factor kf
1.40
1.20
Al-Omaishi (Eq. [22]): kf = 1.2 - 0.05f'ci
AASHTO LRFD specifications (Eq. [12]):
1.23/(0.67 + f'c /9)
1.00
0.80
0.60
0.40
Proposed NCHRP 496 (Eq. [23]): kf = 5/(1 + f'ci)
0.20
2
3
4
5
6
7
8
9
10
11
12
13
14
Compressive strength f'c, ksi
Figure 7. This graph shows the effect of different strength correction factors for different concrete compressive strengths. Note: Assume f'c i = 0.8f'c . AASHTO = American
Association of State Highway and Transportation Officials; f'c i = specified compressive strength of concrete at time of initial loading or prestressing; LRFD = load- and
resistance-factor design; NCHRP = National Cooperative Highway Research Program. 1 ksi = 6.895 MPa.
Time-development correction factor ktd
1.000
Proposed NCHRP 496 (Eq. [25]): ktd = t/(61 - f'ci + t)
0.900
0.800
0.700
Shrinkage
AASHTO LRFD specifications steam curing:
ktd = t/(55 + t) (Eq. [6])
AASHTO LRFD specifications moist curing:
ktd = t/(35 + t) (Eq. [7])
0.600
0.500
Creep
AASHTO LRFD specifications (Eq. [16]):
ktd = (t - ti)0.6/[10 + (t - ti)0.6]
ti = 1
0.400
0.300
Proposed NCHRP 496
0.200
AASHTO LRFD specifications shrinkage (steam)
AASHTO LRFD specifications shrinkage (moist)
0.100
AASHTO LRFD specifications creep
0.000
0
100
200
300
400
500
600
Time t, days
Figure 8. This graph shows the time-development correction factor by various methods. Note: AASHTO = American Association of State Highway and Transportation Officials; f'c i = specified compressive strength of concrete at time of initial loading or prestressing; LRFD = load- and resistance-factor design; NCHRP = National Cooperative
Highway Research Program; t = age of concrete between time of loading for creep calculations or end of curing for shrinkage calculations and time being considered for
analysis of creep or shrinkage effects; ti = age of concrete when load is initially applied.
54
S um me r 2 0 0 9 | PCI Journal
900
800
Predicted ACI 209
700
Predicted AASHTO LRFD
specifications
NCHRP 496 test
data
Shrinkage
600
Proposed NCHRP 496
FHWA showcase
500
ACI 209
400
300
AASHTO LRFD
specifications
200
Proposed
NCHRP 496
100
0
0
20
40
60
80
100
120
140
160
180
200
Time, days
Figure 9. This graph shows the experimental results of shrinkage. Note: AASHTO = American Association of State Highway and Transportation Officials; ACI = American
Concrete Institute; FHWA = Federal Highway Administration; LRFD = load- and resistance-factor design; NCHRP = National Cooperative Highway Research Program.
3.50
3.00
Creep coefficient
2.50
Predicted AASHTO
LRFD specifications
NCHRP 496
test data
Predicted ACI 209
FHWA
showcase
2.00
ACI 209
Proposed NCHRP 496
AASHTO LRFD
specifications
1.50
Proposed
NCHRP 496
1.00
0.50
0.00
0
50
100
150
200
Time, days
Figure 10. This graph shows the experimental results of creep. Note: AASHTO = American Association of State Highway and Transportation Officials; ACI = American Concrete Institute; FHWA = Federal Highway Administration; LRFD = load- and resistance-factor design; NCHRP = National Cooperative Highway Research Program.
of deck placement is to be made. The camber at that time
is used to set girder seating elevations and to determine
concrete haunch size and quantity over the girder and below the deck. This camber is becoming a significant design
parameter with the increased use of HSC and corresponding high levels of prestress.
Figure 8 shows a comparison of the time-development
correction factors calculated with various prediction
methods. The pre-2005 AASHTO LRFD specifications
and ACI 209 use the same time-development correction
factor for predicting the shrinkage of concrete, but they
use different formulas for creep. There are two formulas
for shrinkage depending on type of curing: Eq. (6) and (7).
Recent research presented in PCI’s Precast Prestressed
PCI Journal | S u m m e r 2009
55
Table 3. Ratios of predicted to measured modulus of elasticity of concrete Ec
Ratio of predicted to measured Ec
Coarse aggregate
type
Mixture
AASHTO LRFD
specifications/ACI 318
K1
ACI 363
Proposed
wc = 0.145
kip/ft3
wc = 0.150
kip/ft3
wc = 0.145
kip/ft3
K1 = 1.0
Variable
K1
Nebraska: NE04D, 09G, 10G, 12G, field
Crushed limestone
0.972
0.985
1.037
0.881
1.029
1.0
New Hampshire: NH04D, 10G, 11G, 12G, field
Gravel
0.910
1.066
1.122
0.958
1.099
1.0
Texas: TX04D, 08G, 09G, 10G, field
Crushed limestone
1.299
0.739
0.777
0.650
0.770
1.0
Washington: WA04D, 10G, 11G, 12G, field
Gravel
1.152
0.845
0.889
0.765
0.868
1.0
Average of participating states’ data shown in Fig. 7
0.915
0.963
0.820
0.948
Average of all data, including previous data shown in Fig. 8
0.987
1.037
0.875
1.020
Note: AASHTO = American Association of State Highway and Transportation Officials; ACI = American Concrete Institute; f c' = specified compressive
strength of concrete at 28 days unless another age is specified; K1 = aggregate-stiffness correction factor; LRFD = load- and resistance-factor design;
wc = density of concrete. 1 ft = 0.305 m; 1 lb = 0.453 kg.
Table 4. Ratios of predicted to measured shrinkage and creep coefficient
Creep coefficient
Shrinkage strain
Mixture
ACI 209
AASHTO-LRFD
specifications
Proposed
ACI 209
AASHTO-LRFD
specifications
Proposed
Nebraska: NE04D, NE09G, NE10G, NE12G, NE field
1.75
1.91
1.08
1.69
1.31
1.00
New Hampshire: NH04D, NH10G, NH11G, NH12G, NH field
1.13
1.27
0.80
1.50
1.37
0.84
Texas: TX04D, TX08G, TX09G, TX10G,TX field
2.26
2.60
1.57
2.06
1.89
1.08
Washington: WA04D, WA10G, WA11G, WA12G, WA field
1.05
1.18
0.74
1.89
1.88
0.99
Average of all data
1.55
1.74
1.05
1.79
1.61
0.98
Note: AASHTO = American Association of State Highway and Transportation Officials; ACI = American Concrete Institute; LRFD = load- and resistancefactor design.
Concrete Bridge Design Manual proposes possible modifications to account for concrete strength. As shown in Fig.
9 and 10, development of both shrinkage and creep is more
accelerated at an early age in high-strength than in normalstrength concrete, in which development is more gradual
over a longer period.
The proposed correction factor for time development of
both shrinkage and creep for both conditions of curing is
calculated from Eq. (24).
ktd =
t
61− 4 f ci' + t
(24)
where
t
56
= age of concrete between time of loading for creep
S um me r 2 0 0 9 | PCI Journal
calculations or end of curing for shrinkage calculations and time being considered for analysis of creep
or shrinkage effects
Equation (24) should not be used for concrete strength
at release in excess of 12 ksi (82 MPa) and at service in
excess of 12 ksi/0.8 (83 MPa/0.8) or 15 ksi (103 MPa).
Comparison of experimental
results, prediction methods
Figure 2 shows the experimental results for modulus of
elasticity from this research, while Fig. 3 combines the
results with those from previous research, including those
reported in the ACI 363 committee report and the Federal
Highway Administration (FHWA) showcase projects.23
AASHTO LRFD specifications prediction
800
28%
100%
Predicted shrinkage strain
700
600
500
-28%
400
300
200
NCHRP 496 measured data
100
FHWA showcase data
0
0
100
200
300
400
500
600
700
800
Measured shrinkage strain
ACI 209 prediction
800
28%
Predicted shrinkage strain
700
100%
600
500
-28%
400
300
200
100
NCHRP 496 measured data
FHWA showcase data
0
0
100
200
300
400
500
600
700
800
Measured shrinkage strain
Proposed NCHRP 496 prediction
800
28%
Predicted shrinkage strain
700
100%
600
500
-28%
400
300
200
NCHRP 496 measured data
100
FHWA showcase data
0
0
100
200
300
400
500
600
700
800
Measured shrinkage strain
Figure 11. This graph compares measured with predicted values of shrinkage strain using AASHTO LRFD specifications, the ACI 209 committee report, and the proposed
NCHRP 496 methods. Note: AASHTO = American Association of State Highway and Transportation Officials; ACI = American Concrete Institute; FHWA = Federal Highway
Administration; LRFD = load- and resistance-factor design; NCHRP = National Cooperative Highway Research Program.
PCI Journal | S u m m e r 2009
57
AASHTO LRFD specifications prediction
3.000
100%
28%
Predicted creep coefficient
2.500
2.000
-28%
1.500
1.000
0.500
NCHRP 496 measured data
FHWA showcase data
0.000
0.000
0.500
1.000
1.500
2.000
2.500
3.000
Measured creep coefficient
ACI 209 prediction
3.000
28%
100%
Predicted creep coefficient
2.500
2.000
-28%
1.500
1.000
0.500
!"#$$%&'
$)*+,-./01$2+340,$
NCHRP
496 measured
data
(
FHWA showcase data
0.000
0.000
0.500
1.000
1.500
2.000
2.500
3.000
Measured creep coefficient
NCHRP 496 prediction
3.000
28%
100%
Predicted creep coefficient
2.500
2.000
-28%
1.500
1.000
0.500
NCHRP 496 measured data
FHWA showcase data
0.000
0.000
0.500
1.000
1.500
2.000
2.500
3.000
Measured creep coefficient
Figure 12. This graph compares measured with predicted values of creep coefficient using AASHTO LRFD specifications, the ACI 209 committee report, and proposed
NCHRP 496 methods. Note: AASHTO = American Association of State Highway and Transportation Officials; ACI = American Concrete Institute; FHWA = Federal Highway
Administration; LRFD = load- and resistance-factor design; NCHRP = National Cooperative Highway Research Program.
58
S um me r 2 0 0 9 | PCI Journal
Figures 9 and 10 plot experimental shrinkage and creep
results, respectively. More details are given in NCHRP
report 496.
Table 3 gives a summary of ratios of predicted to measured modulus of elasticity by various methods. It shows
that the average determined by the proposed formula using
all test results is similar to the average determined by the
pre-2005 AASHTO LRFD specifications and the ACI 318
formula. However, significant improvements are obtained
if the aggregate stiffness factor K1 is considered. The
values of K1 shown in Table 3 are only valid for aggregates
similar to the corresponding aggregates used in the project.
Otherwise, local testing must be performed to establish an
appropriate value of K1.
Figure 11 shows the measured versus predicted values of
shrinkage using the pre-2005 AASHTO LRFD specifications, ACI 209, and proposed methods. Also shown in the
figure are 28% upper- and lower-bound standard deviations that correspond to a 95% statistical confidence level.
The figure demonstrates that the new method produces
more-accurate predictions of the average, lower bound, and
upper bound. Figure 12 shows similar results for creep.
Yehia, Nick Meek, Kelvin Lein, and Emil Tadros of UNL,
who provided assistance during the experimental phases of
the project; bridge engineer Lyman Freemon and assistant
bridge engineers Gale Barnhill and Sam Fallaha of the Nebraska Department of Roads and David Scott of the New
Hampshire Department of Transportation, Kevin Pruski
of the Texas Department of Transportation, and Arlen
Clark of Clark County, Wash., who generously offered to
instrument the participating bridges in their states; and Bill
Augustus of Northeast Concrete Products, Robert Steffen
of the University of New Hampshire, Burson Patton of
Texas Concrete Co., Jim Parkins of Concrete Technology,
and Mark Lafferty of Concrete Industries for allowing the
researchers to instrument their products and provide assistance during the laboratory and field-testing program of
the high-strength and normal-strength concrete in Nebraska, New Hampshire, Texas, and Washington.
References
1.Seguirant, S. J. 1998. New Deep WSDOT Standard
Sections Extend Spans of Prestressed Concrete Girders. PCI Journal, V. 43, No. 4 (July–August): pp.
92–119.
Table 4 compares the average ratios of predicted to measured values of shrinkage and creep by various methods. In
general, the proposed method is in closer agreement with
measured data than the other methods are.
2.Stallings, J. M., R. W. Barnes, and S. Eskildsen.
2003. Camber and Prestress Losses in Alabama HPC
Bridge Girders. PCI Journal, V. 48, No. 5 (September–October): pp. 90–104.
Results
3.American Association of State Highway and Transportation Officials (AASHTO). 2004. AASHTO
LRFD Bridge Design Specifications. 3rd ed. Washington, DC: AASHTO.
•
•
•
The proposed formula for modulus of elasticity allows
for variation in coarse aggregate type and stiffness as
well as the effect of increasing density with increased
concrete strength.
The proposed shrinkage-prediction method produced
results that averaged 105% of the measured values, compared with 174% when using the pre-2005
AASHTO LRFD specifications method and 155%
when using the ACI 209 committee-report method.
The proposed creep-prediction method produced results that averaged 98% of the measured values, compared with 161% and 179% for those estimated using
the pre-2005 AASHTO LRFD specifications and the
ACI 209 committee-report methods, respectively.
Acknowledgments
The authors thank Amir Hanna, senior program officer
of the National Cooperative Highway Research Program
on which this summary paper is based; James Gallt, who
provided valuable technical input at the early stage of
this research work; Kromel Hanna and Wilast Pong, who
helped in the preparation of tables and figures; Sharif
4.Tadros, M. K., N. Al-Omaishi, S. Seguirant, and J.
Gallt. 2003. Prestress Losses in Pretensioned HighStrength Concrete Bridge Girders. National Cooperative Highway Research Program (NCHRP) report
no. 496. Washington, DC: Transportation Research
Board, National Academy of Sciences.
5.American Concrete Institute (ACI) Committee 209.
1992. Prediction of Creep, Shrinkage, and Temperature Effects in Concrete Structures. Detroit, MI: ACI.
6.ACI Committee 318. 1999. Building Code Requirements for Structural Concrete (ACI 318-99) and
Commentary (ACI 318R-99). Detroit, MI: ACI.
7.ACI Committee 363. 1992. State of the Art Report on
High-Strength Concrete. Detroit, MI: ACI.
8.Carrasquillo, R. L., A. H. Nilson, and F. O. Slate.
1981. Properties of High Strength Concrete Subject
to Short-Term Loads. ACI Journal, V. 78, No. 3
(May–June): pp. 171–178.
PCI Journal | S u m m e r 2009
59
9.
Russell, H. G. 2002. Personal communication.
10.Shideler, J. J. 1957. Lightweight-Aggregate Concrete
for Structural Use. Journal of the American Concrete
Institute, V. 29, No. 4 (October): pp. 299–328.
11.Myers, J. J., and R. L. Carrasquillo. 1999. The Production and Quality Control of High Performance
Concrete in Texas Bridge Structures. Preliminary
report no. 580-589-1, Center for Transportation Research, Austin, TX.
Nebraska, Lincoln, NE.
23.Federal Highway Administration (FHWA). 2002.
Compilation and Evaluation of Results from High
Performance Concrete Bridge Projects. Contract No.
DTFH61-00-C-00009. Washington, DC: FHWA.
Notation
Ec
= modulus of elasticity of concrete
f c' = specified compressive strength of concrete at 28
days unless another age is specified
fci' = specified compressive strength of concrete at time
of initial loading or prestressing
13.ASTM C39. 2001. Standard Test Method for Compressive Strength of Cylindrical Concrete Specimens.
West Conshohocken, PA: ASTM International.
kc
= size factor for creep
kf
= factor for the effect of concrete strength
14.ASTM C469-94. 1994. Standard Test Method for
Static Modulus of Elasticity and Poisson’s Ratio of
Concrete in Compression. West Conshohocken, PA:
ASTM International.
khc
= humidity factor for creep
khs
= humidity factor for shrinkage
kla
= loading-age correction factor
ks
= size factor
ktd
= time development factor
kvs
= factor for the effect of volume-to-surface ratio
K l
= aggregate stiffness correction factor
= 1.0 unless determined by physical test and as approved by the authority of jurisdiction
RH
= average annual ambient mean relative humidity
t
= age of concrete between time of loading for creep
calculations or end of curing for shrinkage calculations and time being considered for analysis of
creep or shrinkage effects
ti
= age of concrete when load is initially applied
tla
= age of concrete when load is initially applied
12.ASTM C192/C192M. 1992. Standard Practice for
Making and Curing Concrete Test Specimens in the
Laboratory. West Conshohocken, PA: ASTM
International.
15,ASTM C157. 2001. Standard Test Method for Length
Change of Hydraulic Cement Mortar and Concrete.
West Conshohocken, PA: ASTM International.
16.ASTM C 512. 2003. Standard Test Method for Creep
of Concrete in Compression. West Conshohocken,
PA: ASTM International.
17.ASTM C1231/C1231M. 2001. Standard Practice for
Use of Unbonded Caps in Determination of Compressive Strength of Hardened Concrete Cylinders. West
Conshohocken, PA: ASTM International.
18.AASHTO. 2005. AASHTO LRFD Bridge Design
Specifications: 2005 Interim Revisions. Washington,
DC: AASHTO.
19.AASHTO. 2006. AASHTO LRFD Bridge Design
Specifications: 2006 Interim Revisions. Washington,
DC: AASHTO.
20.AASHTO. 2007. AASHTO LRFD Bridge Design
Specifications. 4th ed. Washington, DC: AASHTO.
V/S = volume-to-surface ratio of the member
21.PCI Bridge Design Manual Steering Committee.
1997. Precast Prestressed Concrete Bridge Design
Manual. 1st ed. Chicago, IL: PCI.
22.Al-Omaishi, N. 2001. Prestress Losses in Pretensioned High-Strength Concrete Bridge Girders. PhD
diss., Department of Civil Engineering, University of
60
S um me r 2 0 0 9 | PCI Journal
wc
= density of concrete
εsh
= shrinkage strain
ψ(t,ti)= girder creep coefficient at time t for loading at
time ti
Appendix: Numerical example
This example uses the data of example 9.4 of the Precast
Prestressed Concrete Bridge Design Manual.21 The bridge
consists of 72-in.-deep (1.8 m) AASHTO-PCI bulb-tee
girders spaced at 9 ft (2.7 m). The girders are designed
to act compositely with the 8 in. (200 mm) cast-in-place
concrete deck to resist the superimposed dead loads and
live loads. The superimposed dead loads consist of the
railing and a 2 in. (50 mm) future wearing surface. Both
are assumed for this example to be introduced immediately
after the deck has gained design strength. The cast-in-place
haunch over the girder top flange is assumed to be 0.5 in.
(13 mm) thick and 42 in. (1.1 m) wide.
The bridge is constructed in a region with relative humidity RH of 70%. Precast concrete strength at release fci' is
5.8 ksi and at service f c' is 6.5 ksi. Cast-in-place concrete
compressive strength at 28 days f c' is 4.0 ksi. The aggregate stiffness factor Kl is 1.0. Volume-to-surface ratio
V/S is 3 for the precast concrete girder and 3.51 for the
deck. The construction schedule allows for the following
assumptions:
Creep
Girder
Creep coefficient at final time due to loading at transfer
ψb(tf ,ti)
t = tf – ti = 20,000 – 1 = 19,999 days
kvs = 1.45 – 0.13(V/S) = 1.45 – (0.13)(3) = 1.06 ≥ 0
khc = 1.56 – 0.008RH = 1.56 – (0.008)(70) = 1.00
kf =
ktd =
5
1+
f ci'
=
5
= 0.74
1+ 5.8
( )
t
61 4
f ci'
+t
=
61
19,999
= 1.00
4 5.8 + 19,999
( )( )
( )
ψ b t f ,ti = 1.9kvs khc k f ktd ti−0.118
( )(
Concrete age at prestress transfer ti is 1 day.
)(
)(
)(
)( )−0.118
== (1.9)(1.06)(1.00)(0.74)(1.00)(1)
1.9 1.06 1.00 0.74 1.00 1 -0.118 = 1.48
Girder creep coefficient ψb(td ,ti) at time of deck placement
due to loading introduced at transfer:
Age at deck placement td is 90 days.
Final conditions are assumed to occur at concrete age tf of
20,000 days.
Material properties
td = 90 days, and t = tf – ti = 90 – 1 = 89 days
Modulus of elasticity of concrete:
⎛
⎞
89
t
ktd = ⎜
= 0.70
⎟=
'
5.8 + 89
61−
4
⎝ 61− 4 f ci + t ⎠
Ec = 33,000K1w1.5
f c' c
−0.118
ψ b t df ,ti = 1.9kvs k hc k f ktd kti−0.118
= (1.48)(0.7) = 1.04
i
Girder at release:
= 1.9
1.06 1.00
0.74
Girder creep
coefficient
at final
time1.00
due to1 loading at deck
placement, ti = 90 days
6.5
Ec = 33,000 1.0 0.14 +
1000
(
)( )
( )( )
( )
( )(
1.5
5.8 = 4456 ksi
)(
)(
)(
)( )−0.118
(( )
−0.118
ψψbb ttff ,t,tdi = 1.9kvs khc k f ktd tki−0.118
= (1.48)(90)-0.118 = 0.87
i
( )(
)(
)(
)(
)( )−0.118
= 1.9 1.06 1.00 0.74 1.00 1
Deck
Girder at final time:
6.5
Ec = 33,000 1.0 0.14 +
1000
(
)( )
1.5
6.5 = 4718 ksi
Deck:
kvs = 1.45 – 0.13(V/S) = 1.45 – (0.13)(3.51) = 0.99 ≥ 0
khc = 1.56 – 0.008RH = 1.56 – (0.008)(70) = 1.00
kf =
⎛
4 ⎞
Ec = 33,000 1.0 ⎜ 0.14 +
⎟⎠
1000
⎝
(
)( )
5
1+
f ci'
=
5
= 1.19
1+ 0.80 4
(
)( )
1.5
4 = 3607 ksi
Deck creep at final time due to loads introduced shortly
after deck placement:
PCI Journal | S u m m e r 2009
61
(( )
−0.118
ψψbb ttff ,t,tdi = 1.9kvs khc k f ktd tki i−0.118
( )(
)(
)(
)(
)( )−0.118
1.9 1.06 1.00 0.74 1.00 1–0.118 = 2.24
== (1.9)(0.99)(1.00)(1.19)(1.00)(1)
Shrinkage
Girder
Shrinkage strain between prestress transfer and final time:
khs = (2.00 – 0.014RH) = 2.00 – (0.014)(70) = 1.02
εbif= kvs khs kf ktd 0.48 × 10-3
= (1.06)(1.02)(0.74)(1.00)(0.00048)
= 0.000384 in./in.
Girder shrinkage strain between initial time and deck
placement time, t = 90 – 1 = 89 days:
⎛
⎞
89
t
ktd = ⎜
= 0.70
⎟=
'
⎝ 61− 4 f ci + t ⎠ 61− 4 5.8 + 89
( )( )
εbid = kvs khs kf ktd 0.48 × 10-3 = (0.70)(0.000348)
= 0.000269 in./in.
Girder shrinkage strain between deck placement and final
time:
εbdf = εbif – εbid = 0.000384 – 0.000269 = 0.000115 in./in.
Deck
Shrinkage strain between end of deck curing and final
time:
khs = (2.00 – 0.014RH) = 2.00 – (0.014)(70) = 1.02
εddf = kvs khs kf ktd 0.48 × 10-3 = (0.99)(1.02)(1.19)(1.00)(0.00048)
= 0.000579 in./in.
62
S um me r 2 0 0 9 | PCI Journal
About the authors
Nabil Al-Omaishi, PhD, P.E., is
an associate professor and chair
for the Department of Civil
Engineering at the College of
New Jersey in Ewing, N.J.
Maher K. Tadros, PhD, P.E.,
FPCI, is a Leslie D. Martin
Professor for the Department of
Civil Engineering at the University of Nebraska–Lincoln in
Omaha, Neb.
Stephen J. Seguirant, P.E., is the
vice president and director of
engineering for Concrete Technology Corp. in Tacoma, Wash.
Synopsis
The use of high-strength concrete (HSC) for pretensioned concrete bridge girders has become commonplace among state highway agencies because of its
economic and durability benefits. This paper summarizes part of the research work performed under
the National Cooperative Highway Research Program
(NCHRP) project 18-07, Prestress Losses in Pretensioned High-Strength Concrete Bridge Girders, which
is fully documented in NCHRP report no. 496. The
researchers were assigned the task of extending the
American Association of State and Highway Transportation Officials’ (AASHTO’s) AASHTO LRFD
Bridge Design Specifications provisions for estimating
prestress losses to cover concrete strengths up to 15
ksi (104 MPa).
This paper summarizes the portion of that work on
concrete properties that have an impact on design for
long-term effects: modulus of elasticity, shrinkage,
and creep. These research findings were adopted into
the 2005 and 2006 interim provisions of the AASHTO
LRFD specifications. The experimental component
of the research includes testing of specimens produced from raw materials and mixture proportions
provided by four participating states (Nebraska, New
Hampshire, Texas, and Washington) to encompass the
regional diversity of materials throughout the country.
The theoretical component of the research addresses
the background of prior prediction formulas and the
development of the new formulas that have now been
adopted.
Keywords
Creep, high-strength concrete, HSC, material properties, modulus of elasticity, prestress loss, relaxation,
shrinkage.
Review policy
This paper was reviewed in accordance with the
Precast/Prestressed Concrete Institute’s peer-review
process.
Reader comments
Please address any reader comments to PCI Journal
editor-in-chief Emily Lorenz at elorenz@pci.org or
Precast/Prestressed Concrete Institute, c/o PCI Journal,
209 W. Jackson Blvd., Suite 500, Chicago, IL 60606. J
PCI Journal | S u m m e r 2009
63